Free Cofactor Matrix Calculator

What Is the Cofactor Matrix?

For any square matrix $A$, the cofactor matrix (often called the matrix of cofactors) is obtained by replacing every element $a_{ij}$ with its corresponding cofactor $C_{ij}$. This tool automates the entire process, letting you instantly compute the cofactor matrix for dimensions $2 \times 2$, $3 \times 3$, or larger — all you need are the original entries.

Two building blocks underpin every cofactor: the minor and the sign factor. The minor $M_{ij}$ is the determinant of the submatrix that remains after removing the $i$-th row and $j$-th column of $A$. Multiplying that minor by the sign factor $(-1)^{i+j}$ gives the cofactor:

$$C_{ij} = (-1)^{i+j} \, M_{ij}$$

The sign factor creates a checkerboard pattern that starts with “+” in the top‑left cell and alternates along each row and column. Thanks to this pattern you can quickly determine the sign needed for any position without re‑computing $(-1)^{i+j}$ each time.

How to Find the Cofactor Matrix by Hand

Finding the cofactor matrix manually follows a repetitive but clear routine:

1. Choose the row index $i$ and column index $j$ of the target element.
2. Strike out the $i$-th row and $j$-th column of $A$ to obtain a smaller square submatrix.
3. Calculate the determinant of this submatrix — this is the minor $M_{ij}$.
4. Apply the sign factor $(-1)^{i+j}$ to convert the minor into the cofactor $C_{ij}$.
5. Place $C_{ij}$ into the cofactor matrix at position $(i,j)$.
6. Repeat steps 1–5 for every pair $(i,j)$ from $1$ to $n$.

The resulting array is the full matrix of cofactors. Below we illustrate this for the two most common sizes.

Cofactor of a 2×2 Matrix

Consider a $2 \times 2$ matrix:

$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

Applying the steps yields the following cofactors:

• $C_{11} = d$ (since $(-1)^{1+1}=+1$ and the minor after deleting row 1, col 1 is $d$)
• $C_{12} = -c$
• $C_{21} = -b$
• $C_{22} = a$

Thus the complete cofactor of a 2×2 matrix is:

$$\text{cofactor}(A) = \begin{bmatrix} d & -c \\ -b & a \end{bmatrix}$$

This compact result is one of the quickest ways to obtain the cofactor matrix for a $2 \times 2$ system.

Cofactor of a 3×3 Matrix

For a $3 \times 3$ matrix the same principle applies, but the minors are now $2 \times 2$ determinants. For example, to find the cofactor of entry $a_{11}$:

• Delete row 1 and column 1 → the remaining submatrix is $\begin{bmatrix} e & f \\ h & i \end{bmatrix}$ (using standard labelling).
• Its determinant is $ei - fh$.
• Multiply by sign factor $(+1)$ → $C_{11} = ei - fh$.

Each of the nine positions requires a similar calculation. A cofactor of a 3×3 matrix therefore involves nine individual $2 \times 2$ determinants, which is precisely where an automated calculator saves enormous effort.

How to Use This Cofactor Matrix Calculator

The tool is designed for speed and convenience:

1. Select the size of your matrix ($2 \times 2$, $3 \times 3$, etc.).
2. Enter every coefficient into the corresponding cell.
3. The calculator instantly displays the full matrix of cofactors at the bottom.

There is no need to manually track signs or compute multiple determinants — the underlying cofactor formula is implemented for you.

Relationship to the Adjugate Matrix

The adjugate (or adjoint) matrix is simply the transpose of the cofactor matrix. If you need the adjugate, you can obtain it by taking the cofactor matrix and swapping rows for columns. This makes the adjugate matrix calculator a natural companion to the cofactor tool.

Using the Cofactor Matrix to Invert a Matrix

The cofactor matrix plays a central role in matrix inversion. The inverse of a nonsingular square matrix $A$ is given by:

$$A^{-1} = \frac{1}{\det(A)} \, \bigl(\text{cofactor}(A)\bigr)^{\!T}$$

where $\bigl(\text{cofactor}(A)\bigr)^{T}$ is the adjugate matrix. In practice the steps are:

1. Compute the cofactor matrix of $A$.
2. Transpose it to get the adjugate.
3. Calculate the determinant of $A$.
4. Multiply the adjugate by $1/\det(A)$.

This “cofactor method” is a classic technique that works for any invertible square matrix, and the cofactor matrix calculator makes step 1 effortless.